An integral model based on slender body theory, with applications to curved rigid fibers

TL;DR

This paper introduces a new integral model based on slender body theory for simulating the motion of curved rigid fibers in viscous flow, with a focus on numerical methods, stability, and applications to complex geometries.

Contribution

The paper develops a smooth integral kernel model derived from nonlocal slender body theory, including a convergent numerical method and a fast algorithm for complex fiber geometries.

Findings

The integral operator is proven negative definite in idealized geometries.

Numerical verification confirms the model's accuracy against known ellipsoid models.

The proposed method demonstrates convergence and stability in simulations.

Abstract

We propose a novel integral model describing the motion of curved slender fibers in viscous flow, and develop a numerical method for simulating dynamics of rigid fibers. The model is derived from nonlocal slender body theory (SBT), which approximates flow near the fiber using singular solutions of the Stokes equations integrated along the fiber centerline. In contrast to other models based on (singular) SBT, our model yields a smooth integral kernel which incorporates the (possibly varying) fiber radius naturally. The integral operator is provably negative definite in a non-physical idealized geometry, as expected from PDE theory. This is numerically verified in physically relevant geometries. We propose a convergent numerical method for solving the integral equation and discuss its convergence and stability. The accuracy of the model and method is verified against known models for ellipsoids. Finally, a fast algorithm for computing dynamics of rigid fibers with complex geometries is developed.